I don't think that Clay would have made a difference in 1860-61. But if he or Webster had been alive in 1854, the Kansas-Nebraska Act might not have passed, and that would certainly have been significant in keeping the Whigs alive and preventing or at least delaying the formation of the Republican Party as we know it.
To quote an old soc.history.what-if post of mine:
***
I have been reading Michael Holt's *The Rise and Fall of the American Whig
Party* (Oxford Univeristy Press 1999). I would strongly advise anyone
iterested in the political background of the ACW to read it. (It is
probably the most comprehensive account of the history of any major
American political party.) Holt emphasizes the importance of the
disintegration of the Whigs (as a bi-sectional alternative to the
Democrats) and their replacement by the Republicans as a cause of the Civil
War. Significantly, he states "Next to Tyler's Texas adventurism...no Whig
action did more to destroy the party and bring on the war than southern
Whigs' easily avoidable support for the Nebraska Act in 1854, a mistake
that many of them later rued." pp. 982-3 (yes, it's a long book!)
So what if a few more Southern Whigs had opposed the Kansas-Nebraska Act?
Initially, the Southern Whig press was very hostile to Douglas' proposal.
It would do nothing for the South, they explained--slavery could never
flourish on the Western plains, and "squatter sovereignty" in particular
would guarantee that it would not do so--but would only lead to a reopening
of slavery agitation in a manner very dangerous for the South.
Unfortunately, the first major statement of opposition to the K-N act was
the "Appeal of the Independent Democrats" by the most extreme anti-slavery
men in Congress--Sumner, Chase, etc. Far from claiming that the Union was
being jeopardized for the sake of an abstraction, the "Appeal" argued that
slavery was a real danger in the West. After that, any Southern Whig who
opposed the K-N Act would be jeered at as an "abolitionist."
Yet in spite of this, some Southern Whigs did vote against the K-N Act. On
May 22, the Act passed the House by 113-100. "Southern Whigs split
thirteen for, seven against, and four not voting. One of the abstainees,
Samuel Carruthers of Missouri, missed the vote because he was paired with
an absent northern Whig, and he later announced that he would have
supported passage, as did the three Missouri Whigs who voted. Had half of
those fourteen Southern Whigs--say, Felix Zollicoffer and Charles Ready of
Tennessee, John Kerr of North Carolina, and the four Kentucky Whigs who
voted Yea--instead voted Nay, the bill would have been defeated.." p. 821
Furthermore, there were three abstainers--two from Maryland and one from
Kentucky--who probably opposed the bill even if they did not dare vote
against it. In short, Southern Whigs provided the crucial margin for the
bill, even though 100 of its 113 supporters were Democrats.
What would it have taken to get Upper South Whigs to oppose the bill as
Northern Whigs *unanimously* did? Perhaps if Clay or Webster had lived a
few years longer. They would have emphasized the danger of the K-N Act to
the Compromise of 1850, the "finality" of which was dogma to Southern
Whigs. "You are undermining the pro-finality forces in the North and
thereby endangering the South, the Whig Party, and the Union, and all for
an abstraction" would be their theme. They would demolish Douglas'
sophistry about the Compromise of 1850 having intended to repeal the
Missouri Compromise. They could well have swayed the few needed votes.
https://groups.google.com/d/msg/soc.history.what-if/vcchm0mxuUo/_hMASl--4e4J